In particular, we observe that the superior stable manifold of \(P_2\)  \(W^s_\swarrow P_2\)  connects the boundaries of region \(\Phi \)  defined in Theorem . By continuity of the vector field in \(S\) , it is clear that tthe curve determined by \(W^s_\nearrow \left(1,0\right)\)  remain at \(\Phi \)  by Theorem  and its \(\omega \) -limit can be the point \(P_3\) , see Figure REF . Assuming that, the \(\alpha \) -limit of \(W^s_\swarrow P_2\)  is out of \(\Phi \) , then the curve \(\Sigma \)  is above the curve determined by \(W^s_\nearrow \left(1,0\right)\) . If the \(\alpha \) -limit of \(W^s_sP_2\)  is inside of \(\Phi \) , then the curve \(\Sigma \)  is below the curve determined by \(W^s_\nearrow \left(1,0\right)\) . Then, by By continuity of the vector field in \(S\) , there exists conditions in the \(\left(Q,S\right)\) -parameters space for which the two manifolds \(W^s_\nearrow \left(1,0\right)\)  and \(W^s_\swarrow P_2\)  coincide, forming the heteroclinic curve \(\Sigma \)  [1]}. Furthermore, there exists \(S\)  for which \(W^s_{\swarrow }P_2\)  connects with \(W^u_{\nearrow }P_2\)  (i.e. \(W^s_{\swarrow }P_2\cap W^u_{\nearrow }P_2\) ) generating an homoclinic curve. Then, we have a non–infinitesimal limit cycle which could coincide with other limit cycle created around \(P_2\)  via the Hopf bifurcation and terminates via a homoclinic bifurcation [2]}. On the other hand, there exists conditions in the \(S\) -parameter space for which the equilibrium points \(P_1\)  and \(P_3\)  can be both unstable or one of them can be a stable equilibrium point surrounded by an unstable limit cycle or both can be a stable equilibrium points surrounded by an unstable limit cycle. Hence, the equilibrium points \(P_1\) , \(P_2\)  and \(P_3\)  should be surrounded by a stable limit cycle.
